Category:Sobolev Spaces

This category contains results about Sobolev Spaces.
Definitions specific to this category can be found in Definitions/Sobolev Spaces.

Let $\openint a b$ be an open real interval.

Let $\map {L^2} {a, b}$ be the Lebesgue space.

Let $f \in \map {L^2} {a, b}$.

Suppose that in the sense of distributions we have that:

$f, \ldots, f^{\paren n} \in \map {L^2} {a, b}$

where $n \in \N$.

Then the Sobolev space is defined and denoted as:

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$\map {H^n} {a, b} := \set {f \in \map {L^2} {a, b} : f', \ldots, f^{\paren n} \in \map {L^2} {a, b} }$

and equipped with the Sobolev norm.